A frequency averaging framework for the solution of complex dynamic systems.

Lecomte C - Proc. Math. Phys. Eng. Sci. (2014)

Bottom Line:
A frequency averaging framework is proposed for the solution of complex linear dynamic systems.It is remarkable that, while the mid-frequency region is usually very challenging, a smooth transition from low- through mid- and high-frequency ranges is possible and all ranges can now be considered in a single framework.An interpretation of the frequency averaging in the time domain is presented and it is explained that the average may be evaluated very efficiently in terms of system solutions.

ABSTRACTA frequency averaging framework is proposed for the solution of complex linear dynamic systems. It is remarkable that, while the mid-frequency region is usually very challenging, a smooth transition from low- through mid- and high-frequency ranges is possible and all ranges can now be considered in a single framework. An interpretation of the frequency averaging in the time domain is presented and it is explained that the average may be evaluated very efficiently in terms of system solutions.

RSPA20130743F2: Nominal response of the main mass to a unit force in the (a) frequency and (b) time domains, with and without the additional attachments.

Mentions:
The interest is in the transfer function, , of the main mass, m1, owing to a unit force, f=e1=[1 0 … 0]T, as well as in the impulse time response, , of the same mass. The evaluated corresponding nominal, ‘non-averaged’, functions are presented in figure 2. Some of the difficulties arising when the complexity of the system increases are illustrated on this benchmark problem and now highlighted. First, it is clear that the effect of the additional masses cannot be neglected or simply integrated in the properties of the main system. In the frequency domain, two main parts of the response are identifiable: a single well-isolated resonance in the 6≤ω≤7 range and an apparent blurred resonance pattern made of a combination of several resonances within the range 0≤ω≤5. While the latter roughly looks like a main individual resonance perturbed by a multitude of minor disturbances, such simplified vision is not really the case, since it is really the combination of all individual resonances that creates the blurred pattern. It is also not clear, de visu and qualitatively, if just one or several equivalent blurred resonances are hidden in the pattern. The situation is similarly not that simple in the time domain: the impulse response of the full system exhibits some kind of beating phenomena where its general magnitude appears to first decrease, i.e. be damped, up to about t=50[s] to only increase back up to t=150[s]. Similarly, the response at lower times exhibits a high-frequency, lower magnitude oscillation, that is superposed to a signal with wavelength equal to about 10[s]. Nevertheless, although the impulse response is somewhat complicated, it appears to have some hidden simplified pattern, made of only a small number of components. This situation exhibits several typical characteristics of the problems one is exposed to when working through various frequency ranges. The averaging framework proposed here offers an approach to extract such important components. Before discussing this in more detail, the use of two alternative existing approaches is presented.Figure 2.

RSPA20130743F2: Nominal response of the main mass to a unit force in the (a) frequency and (b) time domains, with and without the additional attachments.

Mentions:
The interest is in the transfer function, , of the main mass, m1, owing to a unit force, f=e1=[1 0 … 0]T, as well as in the impulse time response, , of the same mass. The evaluated corresponding nominal, ‘non-averaged’, functions are presented in figure 2. Some of the difficulties arising when the complexity of the system increases are illustrated on this benchmark problem and now highlighted. First, it is clear that the effect of the additional masses cannot be neglected or simply integrated in the properties of the main system. In the frequency domain, two main parts of the response are identifiable: a single well-isolated resonance in the 6≤ω≤7 range and an apparent blurred resonance pattern made of a combination of several resonances within the range 0≤ω≤5. While the latter roughly looks like a main individual resonance perturbed by a multitude of minor disturbances, such simplified vision is not really the case, since it is really the combination of all individual resonances that creates the blurred pattern. It is also not clear, de visu and qualitatively, if just one or several equivalent blurred resonances are hidden in the pattern. The situation is similarly not that simple in the time domain: the impulse response of the full system exhibits some kind of beating phenomena where its general magnitude appears to first decrease, i.e. be damped, up to about t=50[s] to only increase back up to t=150[s]. Similarly, the response at lower times exhibits a high-frequency, lower magnitude oscillation, that is superposed to a signal with wavelength equal to about 10[s]. Nevertheless, although the impulse response is somewhat complicated, it appears to have some hidden simplified pattern, made of only a small number of components. This situation exhibits several typical characteristics of the problems one is exposed to when working through various frequency ranges. The averaging framework proposed here offers an approach to extract such important components. Before discussing this in more detail, the use of two alternative existing approaches is presented.Figure 2.

Bottom Line:
A frequency averaging framework is proposed for the solution of complex linear dynamic systems.It is remarkable that, while the mid-frequency region is usually very challenging, a smooth transition from low- through mid- and high-frequency ranges is possible and all ranges can now be considered in a single framework.An interpretation of the frequency averaging in the time domain is presented and it is explained that the average may be evaluated very efficiently in terms of system solutions.

ABSTRACTA frequency averaging framework is proposed for the solution of complex linear dynamic systems. It is remarkable that, while the mid-frequency region is usually very challenging, a smooth transition from low- through mid- and high-frequency ranges is possible and all ranges can now be considered in a single framework. An interpretation of the frequency averaging in the time domain is presented and it is explained that the average may be evaluated very efficiently in terms of system solutions.